Can someone first clarify the meaning of 'mining' in the context of TaPOS?
It is not something similar to POW that performs hashing computation.
For TaPOS, I don't think it is hashing or anything computational intensive. That is not the purpose of POS. Is this 'mining' just 'leave the wallet process running'?
Even in 'standard' POS such as Peercoin, there is a "mining" process. This is to create a random/stochastic process for the minting of PoS blocks, similar to the mining of PoW coins. If it were deterministic in a sort of round-robin process, this would create security problems. So there is still a "race" to "discover" the "best" block among all peers.... nobody can guess beforehand who will mint the next PoS block.
Each client works constantly on a hashing problem. The PoS block contains only a single transaction, the miner sending his staked coins to himself. The difficulty is very easy compared to PoW, but it is still a hashing problem. Peercoin keeps the hashrate low by having the hashrate limited to 1 hash per second. The hash is based only on the present timestamp, and some essentially static data (the PoS transaction inputs, and the hashes of previous blocks). So the equivalent to a PoW nonce is the timestamp, which changes only once per second! I suppose somebody could "cheat" by calculating ahead, but the utility of this is limited to the next couple of minutes -- by that time, the next block will be found, which will change the data to be hashed. And the required difficulty decreases according to the number of stake-days included in the block. The stake-days, of course, can be verified by anybody on the network.
So you can still be "lucky" or "unlucky" in PoS mining. There is still a hashing problem, and a difficulty level to be met.
It appears that bytemaster has a different idea in mind for TaPoS mining, which would allow those with greater hashpower to mine blocks more readily. I am not 100% sold on the idea, as it could lead to another hardware arms-race, but the PoS rewards are not likely to be high enough to really matter.
EDIT: It is *necessary* to have mining rewards on top of TaPoS, as simple TaPoS does not require one to participate in the "minting" of blocks in order to obtain one's PoS/"dividends" (unlike Peercoin). Each transaction includes fees which are burned. This increases every stakeholder's proportional interest in the BitShares ecosystem, whether they are connected to the network or not; whether they are mining or not. This greatly simplifies the PoS part, but it requires the addition of mining rewards in order to incentivize mining. If this can be achieved without creating a hardware arms-race, that would be great.